(Ebook) Around the Unit Circle: Mahler Measure, Integer Matrices and Roots of Unity by James McKee, Chris Smyth ISBN 9783030800307, 9783030800314, 303080030X, 3030800318, B09NBHC5CG

(Ebook) Around the Unit Circle: Mahler Measure, Integer Matrices and Roots of Unity by James McKee, Chris Smyth ISBN 9783030800307, 9783030800314, 303080030X, 3030800318, B09NBHC5CG

Mathematics Subject Classification: • 11C08 Polynomials in number theory • 11C20 Matrices, determinants in number theory • 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure • 05A05 Permutations, words, matrices • 05C20 Directed graphs (digraphs), tournaments • 05C22 Signed and weighted graphs • 11R18 Cyclotomic extensions • 11S05 Polynomials • 15B36 Matrices of integers

Mahler measure, a height function for polynomials, is the central theme of this book. It has many interesting properties, obtained by algebraic, analytic and combinatorial methods. It is the subject of several longstanding unsolved questions, such as Lehmer’s Problem (1933) and Boyd’s Conjecture (1981). This book contains a wide range of results on Mahler measure. Some of the results are very recent, such as Dimitrov’s proof of the Schinzel–Zassenhaus Conjecture. Other known results are included with new, streamlined proofs. Robinson’s Conjectures (1965) for cyclotomic integers, and their associated Cassels height function, are also discussed, for the first time in a book.

One way to study algebraic integers is to associate them with combinatorial objects, such as integer matrices. In some of these combinatorial settings the analogues of several notorious open problems have been solved, and the book sets out this recent work. Many Mahler measure results are proved for restricted sets of polynomials, such as for totally real polynomials, and reciprocal polynomials of integer symmetric as well as symmetrizable matrices. For reference, the book includes appendices providing necessary background from algebraic number theory, graph theory, and other prerequisites, along with tables of one- and two-variable integer polynomials with small Mahler measure. All theorems are well motivated and presented in an accessible way. Numerous exercises at various levels are given, including some…